Question 122875
{{{h = -16t^2 + 112t}}} Start with the given equation



{{{180 = -16t^2 + 112t}}} Plug in {{{h=180}}}




{{{16t^2-112t+180=0}}} Get everything to the left side


Let's use the quadratic formula to solve for t:



Starting with the general quadratic


{{{at^2+bt+c=0}}}


the general solution using the quadratic equation is:


{{{t = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}




So lets solve {{{16*t^2-112*t+180=0}}} ( notice {{{a=16}}}, {{{b=-112}}}, and {{{c=180}}})





{{{t = (--112 +- sqrt( (-112)^2-4*16*180 ))/(2*16)}}} Plug in a=16, b=-112, and c=180




{{{t = (112 +- sqrt( (-112)^2-4*16*180 ))/(2*16)}}} Negate -112 to get 112




{{{t = (112 +- sqrt( 12544-4*16*180 ))/(2*16)}}} Square -112 to get 12544  (note: remember when you square -112, you must square the negative as well. This is because {{{(-112)^2=-112*-112=12544}}}.)




{{{t = (112 +- sqrt( 12544+-11520 ))/(2*16)}}} Multiply {{{-4*180*16}}} to get {{{-11520}}}




{{{t = (112 +- sqrt( 1024 ))/(2*16)}}} Combine like terms in the radicand (everything under the square root)




{{{t = (112 +- 32)/(2*16)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{t = (112 +- 32)/32}}} Multiply 2 and 16 to get 32


So now the expression breaks down into two parts


{{{t = (112 + 32)/32}}} or {{{t = (112 - 32)/32}}}


Lets look at the first part:


{{{x=(112 + 32)/32}}}


{{{t=144/32}}} Add the terms in the numerator

{{{t=9/2}}} Divide


So one answer is

{{{t=9/2}}}




Now lets look at the second part:


{{{x=(112 - 32)/32}}}


{{{t=80/32}}} Subtract the terms in the numerator

{{{t=5/2}}} Divide


So another answer is

{{{t=5/2}}}


So our solutions are:

{{{t=9/2}}} or {{{t=5/2}}} (which are {{{t=4.5}}} or {{{t=2.5}}} respectively in decimal form)




So it takes 2.5 seconds for the arrow to climb to 180 ft and then 4.5 seconds for the arrow to fall back to 180 ft